二阶非线性扰动差分方程解的渐近性质

Asymptotic Behavior of Solutions For Nonlinear Perturbations Difference Equations of Second Order

将讨论的差分方程△(rn-1△xn-1)+qnxn=anf(xn)看成是其对应的齐次差分方程△(rn-1△yn-1)+qnyn=0的非线性扰动,其中f(x)为[0,∞)连续函数.设对应的齐次差分方程非振动,zn和yn为其主解和非主解.本文将运用压缩映象原理,获得方程存在渐近于其对应齐次方程主解的解的充分条件,并用方程的系数给出其渐近的精确表示.

Regard the discussed difference equation △（rm-1x0-1）＋qnxn=anf（xn）as nonlinear perturbations of its corresponding homogeneous equation Δ（rn-1Δyn-1）＋qnyn=0 was a continous funcfion on 1-0, ∞）. Under the assumption that the corresponding homogeneous equation is nonoscillatory, let z., y. be principal and nonprincipal solutions of the homogeneous equation respectively. In this paper, we obtain some sufficient conditions and give solutions of a equation asymptotic to principal solution of its homogeneous equation with contraction mapping theorem, then give its accurate expression with coefficients of the equation.